Resonances and Spectral Shift Function near the Landau levels

Jean-François Bony; Vincent Bruneau; Georgi Raikov

Displaying similar documents to “Resonances and Spectral Shift Function near the Landau levels”

Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances

Jean-François Bony, Setsuro Fujiié, Thierry Ramond, Maher Zerzeri (2011)

Annales de l’institut Fourier

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We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of $h$ , and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually,...

Fokker-Planck equation in bounded domain

Laurent Chupin (2010)

Annales de l’institut Fourier

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We study the existence and the uniqueness of a solution $ϕ$ to the linear Fokker-Planck equation $- Δ ϕ + div (ϕ F) = f$ in a bounded domain of $ℝ^{d}$ when $F$ is a “confinement” vector field. This field acting for instance like the inverse of the distance to the boundary. An illustration of the obtained results is given within the framework of fluid mechanics and polymer flows.

A note on the Hermite–Rankin constant

Kazuomi Sawatani, Takao Watanabe, Kenji Okuda (2010)

Journal de Théorie des Nombres de Bordeaux

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We generalize Poor and Yuen’s inequality to the Hermite–Rankin constant $γ_{n, k}$ and the Bergé–Martinet constant $γ_{n, k}^{'}$ . Moreover, we determine explicit values of some low- dimensional Hermite–Rankin and Bergé–Martinet constants by applying Rankin’s inequality and some inequalities proven by Bergé and Martinet to explicit values of $γ_{5}^{'}, γ_{7}^{'}$ , $γ_{4, 2}$ and $γ_{n}$ ( $n ≦ 8$ ).

Semiclassical resolvent estimates at trapped sets

Kiril Datchev, András Vasy (2012)

Annales de l’institut Fourier

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We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs $χ$ microlocally supported away from the trapping: $∥ χ R_{h} (E + i 0) χ ∥ = 𝒪 (h^{- 1})$ , a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, $\tilde{χ}$ , to be supported at the trapped set, giving $∥ χ R_{h} (E + i 0) \tilde{χ} ∥ = 𝒪 (\sqrt{a (h)} h^{- 1})$ when the a priori bound...

On harmonic continuation of differentiable functions defined on a part of the boundary.

Yarmukhamedov, Sharof (2002)

Sibirskij Matematicheskij Zhurnal

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Microlocalization of resonant states and estimates of the residue of the scattering amplitude

Jean-François Bony, Laurent Michel (2003)

Journées équations aux dérivées partielles

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We obtain some microlocal estimates of the resonant states associated to a resonance $z_{0}$ of an $h$ -differential operator. More precisely, we show that the normalized resonant states are $𝒪 (\sqrt{| Im z_{0} | / h}$ $+ h^{\infty})$ outside the set of trapped trajectories and are $𝒪 (h^{\infty})$ in the incoming area of the phase space. As an application, we show that the residue of the scattering amplitude of a Schrödinger operator is small in some directions under an estimate of the norm of the spectral projector. Finally we prove...